EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 783-787
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Bessel type transform associated with Titchmarsh’s
Theorem

Balasaheb Bhagaji Waphare1,∗, Yashodha Sanjay Sindhe1

1 Mathematics Department, MAEER’s MIT Arts, Commerce and Science College, Alandi,
Pune-412105,Maharashtra, India

Abstract. In this paper we have extended Titchmarsh’s theorem for the Bessel transform for
function on half-line [0,∞) in a weighted Lp metric and is studied with the use of Bessel generalized
translation.

2020 Mathematics Subject Classifications: 42A38, 42B37

Key Words and Phrases: Bessel operator, Bessel transform, Bessel generalized translation

1. Introduction and Preliminaries

In recent past, integral transforms are widely used to solve various problems in calculus,
mechanics, Mathematical physics, engineering and computational mathematics (see [4, 6]
the paper by R. Daher, M. EI Hamma and A. EI Houasni, Titehmarsh theorem for the
Bessel Transform, MATEMATIKA, 2012, Vol.28, No.2, 127-131, motivated us to prepare
this paper .
Titchmarsh ([2], Theorem 84) characterized the set of functions in Lp(R) satisfying the
estimate given in the following theorem.

Theorem 1. Let f(x) ∈ Lp(R)(1 < p ≤ 2), and let∫ ∞
−∞
|f(x+ h)− f(x− h)|pdx = O (hαp) (0 < α ≤ 1)

as h→ 0. Then F(f)(x) ∈ Lβ(R) for
p

p+αp−1 < β < p
p−1 , where F(f) stands for the Fourier transform of f.

The main objective of this paper is to establish an analog of Theorem 1 in the Bessel
type operators setting by means of the Bessel generalized translation. Let ∆ = ∆a,b =

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3986

Email addresses: balasahebwaphare@gmail.com (B.B.Waphare), ysindhe@gmail.com (Y.S. Sindhe)

http://www.ejpam.com 783 © 2021 EJPAM All rights reserved.



B.B.Waphare, Y.S. Sindhe / Eur. J. Pure Appl. Math, 14 (3) (2021), 783-787 784

Dtt + a−b
t Dt, be the Bessel type differential operator, where Dt = d

dt . By ja−b−1
2

(t) denote

the Bessel normed function of the first kind. i.e.

ja−b−1
2

(t) =
2
a−b−1

2 Γ
(
a−b+1

2

)
Ja−b−1

2
(t)

t
a−b−1

2

,

where Jν is Bessel function of the first kind with ν = a−b−1
2 and Γ(x) is the Gamma

function (see [1]). The function y = ja−b−1
2

(t) satisfies the differential equation ∆y + y =

0 with the initial conditions y(0) = 1, y′(0) = 0. The function ja−b−1
2

(t) is infinitely

differentiable, entire analytic.
Let Lpa,b(R+), (a− b) > 0 and 1 < p ≤ 2 be the Banach space of measurable functions f(t)
on R+ with the finite norm.

‖f‖ = ‖f‖p,a,b =

(∫ ∞
0
|f(t)|pta−bdt

)1/p

.

Consider the Bessel generalized translation Th in Lpa,b(R+) (see [[4],p.121])

Thf(x) =
Γ
(
a−b+1

2

)
Γ(12)Γ

(
a−b
2

) ∫ π

0
f
(√

x2 + h2 − 2xh cos t
)

sina−b−1 tdt, a− b > 0, 0 ≤ h < 1

which corresponds to the Bessel operator ∆a,b It is not very difficult to see that

T0f(x) = f(x).

If f(x) has a continuous first derivative, then

∂

∂h
Thf(x)|h=0 = 0.

If it has a continuous second derivative, then u(x, h) = Thf(x) solves the Cauchy problem

∂2u

∂x2
+
a− b
x

∂u

∂x
=
∂2u

∂h2
+
a− b
h

∂u

∂h

and

u|h=0 = f(x),
∂u

∂h
|h=0 = 0.

The operator Th is linear, homogeneous and continuous. Below are some properties of this
operator (see [[4], pp.124-125]):
(i) Thja−b−1

2
(λx) = ja−b−1

2
(λh)ja−b−1

2
(λx)

(ii) Th is self-adjoint. If f(x) is continuous function such that
∫∞
0 xa−b|f(x)|dx < ∞, and

g(x) is continuous and bounded for all x ≥ 0 then∫ ∞
0

(Thf(x))g(x)xa−bdx =

∫ ∞
0

f(x)(Thg(x))xa−bdx



B.B.Waphare, Y.S. Sindhe / Eur. J. Pure Appl. Math, 14 (3) (2021), 783-787 785

(iii) Thf(x) = Txf(h)
(iv) ‖Thf − f‖ → 0 as h→ 0.
The Bessel transform defined by the formula (see [1, 3, 4])

f̂(λ) =

∫ ∞
0

f(t)ja−b−1
2

(λt)ta−bdt, λ ∈ R+.

The inverse Bessel transform is given by the formula

f(t) =

(
2
a−b−1

2 Γ

(
a− b+ 1

2

))−2 ∫ ∞
0

f̂(λ)ja−b−1
2

(λt)λa−bdλ.

The following relation connect the Bessel generalized translation, and the Bessel transform
in [5], We have (

T̂hf
)

(λ) = ja−b−1
2

(λh)f̂(λ). (1)

For (a − b) > 0 , we introduce the Bessel normalized function of the first kind ja−b−1
2

defined by

ja−b−1
2

(x) = Γ

(
a− b+ 1

2

) ∞∑
n=0

(−1)n(x/2)2n

n!Γ
(
n+ a−b+1

2

) . (2)

Therefore from (2), we have

lim
x→0

(
ja−b−1

2
(x)− 1

)
x2

6= 0.

By consequence, there exist C > 0 and η > 0 satisfying

|x| ≤ η ⇒ |ja−b−1
2

(x)− 1| ≥ C|x|2. (3)

2. An Analog of Titchmarsh’s Theorem

In this section we give an analog of Titchmarsh’s Theorem 1 (see [[2], Theorem 84])
for the Bessel transform.

Theorem 2. Let f(x) ∈ Lpa,b(R+), (1 < p ≤ 2), and let∫ ∞
0
|Thf(x)− f(x)|pxa−bdx = O(hγp)(0 ≤ γ ≤ 2)

as h→ 0. Then f̂(x) ∈ Lβa,b(R+), for p(a−b+1)
(a−b+1)(p−1)+γp < β ≤ p

p−1 , where 0 < γ < 1

Proof. For a fixed h the Bessel transform of Thf(x) is ja−b−1
2

(hx)f̂(x).

Hence the Bessel transform of Thf(x)− f(x), as a function of x is (ja−b−1
2

(hx)− 1)f̂(x).

Thus∫ ∞
0
|ja−b−1

2
(hx)−1|p′ |f̂(x)|p′xa−bdx < k(p)

(∫ ∞
0
|Thf(x)− f(x)|pxa−bdx

)1/(p−1)
< k(p)hγp

′



B.B.Waphare, Y.S. Sindhe / Eur. J. Pure Appl. Math, 14 (3) (2021), 783-787 786

Now from (3), we have∫ η/h

0
|hx|2p′ |f̂(x)|p′xa−bdx < k(p)hγp

′
, where p′ =

p

p− 1

Then ∫ η/h

0
x2p

′ |f̂(x)|p′xa−bdx < k(p)h(γ−2)p
′
,

Let

ϕ(ξ) =

∫ ξ

1
|x2f̂(x)|βx(a−b)p′/βdx.

Then, if β < p′

ϕ(ξ) ≤
(∫ ξ

1
|x2f̂(x)|p′xa−bdx

)β/p′ (∫ ξ

1
dx

)1−β/p′

= O

(
ξ
(2−γ)p′ β

p′ ξ
1− β

p′

)
= O

(
ξ
2β−γβ+1− β

p′

)
Hence∫ ξ

1
|f̂(ξ)|βxa−bdx =

∫ ξ

1
x
−2β−(a−b) β

p′ ϕ′(x)xa−bdx

= ξ
−2β−(a−b) β

p′ ξa−bϕ(ξ) + (2β + a− b) β
p′
− (a− b)

∫ ξ

1
x
−2β−(a−b) β

p′+(a−b−1)
ϕ(x)dx

= O

(
ξ
−2β−(a−b) β

p′+a−b+1−γβ+β
(
p+1
p

))
+O

(∫ ∞
1

x
−2β−(a−b) β

p′+(a−b−1)
x
1−γβ+β

(
p+1
p

)
dx

)
= O

(
ξ
−2β−(a−b) β

p′+a−b+1−γβ+β
(
p+1
p

))
and this is bounded as ξ →∞ if −2β − (a− b) βp′ + a− b+ 1− γβ + β

(
p+1
p

)
< 0

i.e. if

β >
p(a− b+ 1)

(a− b+ 1)(p− 1) + γp
.

Thus theorem is proved.

3. Conclusion

There are many theorems known about to classical Fourier transform can be generalized
for the Bessel type transform, among them is Titchmarsh’s theorem. We have succussfully
generelised this Titchmarsh’s Theorem for the Bessel transform in this space Lpa,bR+

Remarks:



REFERENCES 787

1. If we take a = α+ 3
4 , b = −α− 1

4 throughout this paper then we obtain the results
studied by R. Daher, M. EI Hamma and A.EI Houashi, published in Matematika,
2012, Vol. 28, No.2, 127-131.

2. Authors claim that results studied in this paper are stronger than that of R. Daher,
M.EI Hamma and A.EI Houasni.

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[3] Trimeche K. Generalized Harmonic Analysis and Wavelet packets. Amsterdam: Gor-
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[4] Kipriyanov I. V. singular Elliptic Boundary value problems. MOSCOW:Nauka., 1997.

[5] Abilov V.A., Abilova F.V., and Kerimov M.K. Estimates for the fourier- bessel integral
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